/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

Toeplitz and circulant matrices

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

How big can the commutator of two matrices be and how big is it typically

One of the great challenges of the spectral theory of singular integral operators is a theory unifying the three "forces" which determine the local spectra: the oscillation of the Carleson curve, the oscillation of the Muckenhoupt weight, and the oscillation of the coecients. In this paper we demonstrate how by employing the method of limit operators one can describe the spectra in case all data of the operator (the curve, the weight, and the coecients) are slowly oscillating.

The method of limit operators for one-dimensional singular integrals with slowly oscillating data

The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range and X and Y are Hilbert spaces. The solution x=x † is assumed to be in the range of some selfadjoint strictly positive bounded linear operator G:X →X. Under several assumptions on G, such as or more generally , inequalities of the form , or range inclusions , convergence rates for the regularization error of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The article contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.

https://www.tu-chemnitz.de/mathematik/preprint/2005/PREPRINT_09.pdf

Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

Toeplitz Operators and Determinants Generated by Symbols with One Fisher‐Hartwig Singularity

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.

Albrecht Böttcher

Papers

How big can the commutator of two matrices be and how big is it typically

The method of limit operators for one-dimensional singular integrals with slowly oscillating data

Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

Toeplitz Operators and Determinants Generated by Symbols with One Fisher‐Hartwig Singularity

From Toeplitz Eigenvalues through Green’s Kernels to Higher-order Wirtinger-Sobolev Inequalities